On The Adjacent Vertex Distinguishing Total Coloring Of 1-Tree

In the litereture [2], Zhang Zhongfu introduced the conception of adjacent vertex distinguishing total coloring. The conception is: Let G be a connected simple graph on at least 2 vertexs, k be a positive integer, f be a mapping from V{G) U E(G) to {1,2, ...,k}. Let C(u) = {f(u)} U {f(uv) | uv ∈ E(G), v ∈ V(G)} for every u ∈ V(G). If(i) for every uv, vw G E(G), u ≠ w, we have f(uv)≠ f(vw);(ii)for every uv ∈ E(G), we have f(u)≠ f(v), f(u) ≠ f(uv), f(v) ≠ f(uv), then f is called a k proper total coloring of G. Furthermore, if f satisfies(iii)For every uv G E(G), we have C(u) ≠ C(v)then f is called a k adjacent vertex distinguishing total coloring of G ( k-AVDTC for short ). The number min{k | G has a k adjacent vertex distinguishing total coloring} is called the adjacent vertex distinguishing total chromatic number and denoted by Xat(G), where C(u) is the set of the colors of it and edges which is adjacent to u. In this thesis, we studied the adjacent vertex distinguishing total coloring of 1-tree and the adjacent vertex distinguishing total chromatic number of 1-tree was given.Some terminology, notation and basic results in the thesis are given in Chapter 1. In Chapter 2, we study the adjacent vertex distinguishing total coloring of 1-cycle and the adjacent vertex distinguishing total chromatic number of 1-cycle is given. The adjacent vertex distinguishing total coloring of 1-tree is talked about in Chapter 3. we give the adjacent vertex distinguishing total chromatic number of 1-tree with Δ(G) ≤ 3 in Section 3.1. Using another basis of induction in the proof, we give the adjacent vertex distinguishing total chromatic number of 1-tree with Δ(G) ≥ 4 in Section 3.2. The main results are listed as following:Theorem Let G be a 1-cycle graph. We haveTheorem Let G be a 1-tree graph. We have...